12x12 Queens Puzzle #32
Difficulty: hard. Place 12 queens on this 12x12 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 29-step reasoning is written out below.
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Step-by-step solution for puzzle #32
What follows is a complete solve of puzzle #32, worked the way a person works it rather than the way a computer does. It runs to 29 deductions — 12 queen placements and 17 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the proof by contradiction.
Techniques this board needs: last square in a region, region shadow, pairing argument on rows, pairing argument on columns, region confined to one line, proof by contradiction, line confined to one region, last square in a row.
Show the full walkthrough (21 steps) — contains the solution
Step 1. Only row 12, column 1 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 1, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 2, the remaining pale yellow squares, and the ring of neighbours around row 7, column 2.
Step 3. The slate region has 4 squares left, spread across rows 4–6, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 4. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Sea green is down to 3 candidates in rows 10 and 11, columns 5 and 6, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about sea green itself.
The board after step 5 — 2 of 12 queens placed. Step 6. The teal region has 5 squares left, spread across rows 9–11, columns 9–11. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 7. Orange, sea green, and teal are now boxed into rows 9–11 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 12 squares at once.
Step 8. Here is the pairing argument. The indigo, mint, orange, and teal regions have no open squares outside columns 9–12 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.
Step 9. Slate cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 6 squares in all.
Step 10. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Blue, red, and sea green are now boxed into columns 5–7 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 10 squares at once.
The board after step 11 — 2 of 12 queens placed. Step 12. The khaki region has 3 squares left, spread across rows 5, 6, and 8, columns 3 and 4. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 13. Test row 1, column 9 for indigo: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. Every open square left in column 9 (2 open squares) belongs to the teal region. Since that column must contain a queen, teal is the colour that supplies it — so teal squares anywhere else on the board are out, 3 squares in all.
Step 15. Look along row 9: one square survives, row 9, column 12, which belongs to orange. A row cannot be left empty, so the queen goes there. That clears row 9, column 12, the rest of the orange region and every square adjacent to it — diagonals included.
Step 16. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 16 — 3 of 12 queens placed. Step 17. Row 2 has only amber squares left (2 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 5 squares.
Step 18. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 10. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 19. Indigo cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 3 squares in all.
Step 20. Now a short proof by contradiction. Suppose the indigo queen went to row 4, column 11. Follow the forced moves that would create and row 1 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 21. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 11; blue has one square left, so its queen takes row 3, column 7; slate has one square left, so its queen takes row 5, column 8; red has one square left, so its queen takes row 6, column 6; khaki has one square left, so its queen takes row 8, column 4; amber has one square left, so its queen takes row 2, column 3; mint has one square left, so its queen takes row 4, column 10; sea green has one square left, so its queen takes row 10, column 5; teal has one square left, so its queen takes row 11, column 9. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 11, column 9.
The board after step 21 — 12 of 12 queens placed.
That completes the board: 12 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 12x12 Queens puzzle #32
This is puzzle number 32 in the 12x12 collection, rated hard. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #32 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #32 are: rose (1 square), slate (5 squares), orange (5 squares), sea green (5 squares), pale yellow (6 squares), indigo (8 squares), blue (10 squares), teal (12 squares), mint (17 squares), red (20 squares), amber (25 squares), khaki (30 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 30 squares.
Every color takes one queen and one only, whatever its size — which is why the 29 square difference between rose and khaki is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Square count is only half the story — how a color is arranged counts too. Rose is the tidiest shape on #32, packing 100% of its 1×1 bounding box, whereas teal straggles across its 4×6 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #32, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Rose occupies 1 square within rows 12–12, columns 1–1. It never leaves row 12, so its queen is locked to that row.
Slate: 5 squares across rows 4–7, columns 8–9.
Orange covers 5 squares, spanning 3 rows and 3 columns (rows 9–11, columns 10–12).
Sea green occupies 5 squares within rows 10–12, columns 5–6.
Pale yellow: 6 squares across rows 4–8, columns 1–2.
Indigo covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 9–12).
Blue occupies 10 squares within rows 1–3, columns 5–10.
Teal: 12 squares across rows 9–12, columns 7–12.
Mint covers 17 squares, spanning 7 rows and 4 columns (rows 2–8, columns 9–12).
Red occupies 20 squares within rows 4–11, columns 5–9.
Amber: 25 squares across rows 1–5, columns 1–7.
Khaki covers 30 squares, spanning 8 rows and 6 columns (rows 5–12, columns 1–6).
Where to start on this board
The most constrained color is where a solve should begin, and on #32 that is rose with 1 square. It reaches from row 12 to row 12 and from column 1 to column 1 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: pale yellow, slate and sea green span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #32
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 3 and 4 are the tightest columns, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 5 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
With twelve queens to seat and twelve regions competing over 144 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 12x12 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #32 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #32 is rated hard
The rating reflects how much the board concedes up front. Puzzle #32 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 29 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 12x12
At 12x12 you are managing 144 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #32 is column 3.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 12x12 Queens puzzle #32?
Puzzle #32 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #32 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #32, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #32?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.